Rank = 1, n = 6

"Morphs" are what we call the shape of the graph, and it turns out that even though there are many thousands of graphs, when their characteristic polynomial is found, many are shared. In fact, some morphs have thousands of different graphs that produce the same morph, but each of those matrices shares the same characteristic polynomial. The characteristic polynomial and it's roots represent the "shape" of the structure, and each graph is a different permutation of numbering of the vertices.

Here are all currently identified morphs, and how many there are, with an example of each one:

There are 15 of

C2,C2,C1

Partial Probability: 2%

λ⁶-3λ⁴+3λ²-1 = 0

There are 40 of

C3,C3

Partial Probability: 6%

λ⁶-2λ³+1 = 0

There are 120 of

C6

Partial Probability: 17%

λ⁶-1 = 0

There are 90 of

C4,C2

Partial Probability: 13%

λ⁶-λ⁴-λ²+1 = 0

There are 120 of

C3,C2,C1

Partial Probability: 17%

λ⁶-λ⁵-λ⁴+λ²+λ-1 = 0

There are 144 of

C5,C1

Partial Probability: 20%

λ⁶-λ⁵-λ+1 = 0

There are 45 of

D2,D2,I2

Partial Probability: 6%

λ⁶-2λ⁵-λ⁴+4λ³-λ²-2λ+1 = 0

There are 90 of

C4,I2

Partial Probability: 13%

λ⁶-2λ⁵+λ⁴-λ²+2λ-1 = 0

There are 40 of

C3,I3

Partial Probability: 6%

λ⁶-3λ⁵+3λ⁴-2λ³+3λ²-3λ+1 = 0

There are 15 of

D2,I4

Partial Probability: 2%

λ⁶-4λ⁵+5λ⁴-5λ²+4λ-1 = 0

There are 1 of

I6

Partial Probability: 0%

λ⁶-6λ⁵+15λ⁴-20λ³+15λ²-6λ+1 = 0